Optimal. Leaf size=103 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {398} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin {align*} \int \frac {1}{\left (-2 a-b x^2\right ) \sqrt [4]{-a-b x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 168, normalized size = 1.63 \[ -\frac {6 a x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )}{\sqrt [4]{-a-b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {b x^2}{a},-\frac {b x^2}{2 a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 56.41, size = 350, normalized size = 3.40 \[ -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {-b x^{2} - a} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )} \sqrt {-a b \sqrt {\frac {1}{a^{3} b^{2}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}-2 a \right ) \left (-b \,x^{2}-a \right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{{\left (-b\,x^2-a\right )}^{1/4}\,\left (b\,x^2+2\,a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{2 a \sqrt [4]{- a - b x^{2}} + b x^{2} \sqrt [4]{- a - b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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